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Questions tagged [statistical-mechanics]

The study of large, complicated systems by means of statistics and probability theory, in order to extract average properties and to provide a connection between mechanics and thermodynamics.

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1answer
30 views

Thermodynamic Entropy seems to be contradictory

For an ideal gas the entropy change with energy is inversely proportional to temperature: This must yield: $$S=\frac 2 3 k_B \ln(T)$$ For various reasons, this equation is hard to find. However ...
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0answers
69 views

What are enthalpic and entropic forces?

Am I right when thinking of entropic force to be an entropy minimizing mechanism and enthalpic force to be an energy minimizing mechanism (which is basically an entropy maximization mechanism). What'...
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2answers
48 views

Question on the temperature dependence of the partition function

Let's just say we're looking at the classical continuous canonical ensemble of a harmonic oscillator, where: $$H = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 x^2$$ and the partition function (...
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1answer
34 views

General formula for the variation of the chemical potential with temperature

For small temperatures $T$, such that $k_BT\ll \mu(T=0)\equiv \mu(0)$, the variation of chemical potential with temperature is given by $$\mu(T)=\mu(0)\Big[1-\frac{\pi^2}{12}\Big(\frac{k_BT}{\mu(0)}\...
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0answers
10 views

$p$-spin spherical spin glass

Consider the $p$-spin spherical spin glass model with Hamiltonian $$H_{N,p}(\sigma)=\frac{1}{{N}^{\frac{(p-1)}{2}}} \sum \limits_{i_1,...i_p} J_{i_1,...i_p} \sigma_{i_1} \sigma_{i_2} .. \sigma_{i_p} $$...
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1answer
15 views

What is the sign of chemical potential of a noninteracting classical ideal gas obeying MB distribution?

The chemical potential of a noninteracting Bose gas can never be negative while that of a noninteracting Fermi gas can be both positive or negative. What can be said about the chemical potential of ...
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2answers
2k views

How do I calculate the probability that the oscillator is in a certain state using partition function?

So let's say I have a single ($N=1$) quantum harmonic oscillator and the energy is determined by $E_n = (n + 1/2) \cdot \hbar \omega$ (where $n$ is the quantum number and $n$ = $0, 1, 2, \ldots$) ...
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How is quantum mechanics consistent with statistical mechanics?

Let's say we have an harmonic oscillator (at Temperature $T$) in a superposition of state 1 and 2: $$\Psi = \frac{\phi_1+\phi_2}{\sqrt{2}}$$ where each $\phi_i$ has energy $E_i \, .$ The probability ...
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3answers
109 views

Conservation of energy and realm of possibility

The law of conservation of energy states that energy cannot be created or destroyed. Based on this principle, you can safely conclude that any effect resulting from a cause must somehow keep all ...
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2answers
214 views

Why do velocities obey the Boltzmann distribution?

So I know the Boltzmann distribution is: $$ P\propto \exp \left(-E / k_BT \right) $$ where $E$ is energy, $k_B$ is the Boltzmann constant and $T$ is the temperature. However, when we replace $E$ for ...
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0answers
34 views

Math problem in Kaufmann-Onsager exact solution to 2D Ising model [on hold]

So, I've been following Huang book in Statistical mechanics for the 2D exact solution to the Ising model (chapter 15). During the solution he has to solve an eigenvalue problem, that is: $$(A+z_kB+...
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2answers
49 views

What happens when we cool down the gas of non-identical particles?

For gas of identical particles, when we cool it down to extremely low temperature we can see one of two types of behaviour depending on the symmetry of wavefunction with respect to argument ...
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1answer
198 views

Aren't the definitions of zero point energy and the 3rd law of the thermodynamics contradictory?

I was listening to a statistical mechanics lecture, and my professor started talking about zero point energy. He defined it as the energy a particle has at 0 K. Doesn't this violate the 3rd law of the ...
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0answers
21 views

Conceptual meaning of Thermal States

Thermal states are generally defined as $$\tau(\beta)= \frac{e^{-\beta H}}{\mathrm{Tr}(e^{-\beta H})}$$ What are some physical statements one can make about them? A system in thermal equilibrium is ...
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2answers
313 views

Thermal/finite temperature quantum field theory: online lectures and best books

There is nice theme about online lectures on QFT. I would like to know about any online lectures on thermal/finite temperature QFT. Also I would like to know about best books on thermal/finite ...
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2answers
403 views

Internal energy of a reversible adiabatic process

what happens to the heat transfer and the internal energy in a system that undergoes reversible adiabatic process back to its original state. Bearing in mind that heat $Q$ was transferred at the ...
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2answers
30 views

How to see Planck's radiation law as a consequence of Bose Einstein statistics?

Planck's law comes about from the following ingredients. 1) The mode density per unit volume in a cavity is $8\pi\nu^2/c^3$. 2) Within each mode, assume Boltzmann statistics i.e the probability of ...
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0answers
23 views

Why do we take mean occupation number of particles? [on hold]

In statistical mechanics, when we find occupation number of particles using Bose-Einstein distribution or Fermi-Dirac distribution, why do we take the mean value of the occupation number? What does it ...
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0answers
18 views

What is the compressibility of this simple “book”? [migrated]

Compressibility is defined as $$C=\frac{2^{HN}}{2^{H_{max}N}}$$ The book is made up of a simple alphabet of only {a,b,c,d} which occur with probabilities $$P(a)=0.2, P(b)=0.4, P(c)=0.1, P(d)=0.3$$ ...
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1answer
207 views

Degeneracy in Maxwell Boltzmann distribution

While deriving Maxwell-Boltzmann distribution function we consider particles having degeneracy. But in classical mechanics how the concept of degeneracy of particle comes within?
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0answers
21 views

The microcanonical ensemble approach to calculating the entropy of an ideal gas [duplicate]

I would like to set up the following problem. Assume I have a box of volume $V$ with $N$ noninteracting particles in it. The energy of each particle can be $\mathcal{E}_i$ such that $\sum_i \mathcal{E}...
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0answers
20 views

What is effective mean value? [on hold]

For counting the number of particles in any specific state we use effective mean value $\langle \hat{N} \rangle $ instead of number operator $\hat{N}$ in ensemble. I want to know what is the advantage ...
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1answer
44 views

Boltzmann distribution derivation from maximum entropy principle

I'm stuck halfway through a derivation of the Boltzmann distribution using the principle of maximum entropy. Let us consider a particle that may occupy any discrete energy level $\mathcal{E}_i$. The ...
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2answers
33 views

How to write equation of state in terms of partition function?

While studying quantum gases (fermions, bosons ), equation of state written were $PV = k_B T Z_{gr}$, where $Z_{gr}$ is the partition function of grand canonical ensemble. P and V are pressure and ...
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1answer
35 views

Increase of entropy as statistical necessity via Fundamental Assumption of Statistical Mechanics

My statistical physics books reasons that the increase of entropy for a closed system arises naturally from statistics. Outline: 1) Fundamental Assumption of Statistical Mechanics: For a system at ...
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1answer
26 views

Probability of a system in the canonical ensemble

In the canonical ensemble, we have the state of system $x_s$ and the state of the environment $x_e$. The probability of the total system is $$P(x_s,x_e)= const.$$ and that is independent of the states ...
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2answers
22 views

Whats the cross sectional area related to shear stress in torsion of a bar?

In shear force in a rectangular bar,the relevant area is the cross sectional area parallel to the applied force.But in torsion which also undergo shearing we get shear stress from torsion equation.I ...
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0answers
17 views

Detailed Balance Violation and Fokker-Planck Equation

Suppose I have a system with N sites, and each site can be modified (M) or anti-modified (A). Transitions between these two states are in part random, and in part auto-regulated by recruitment of At ...
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1answer
32 views

Viscoelastic Constitutive Relation

In the Mori-Zwanzig formalism, the following identification for the generalised shear viscosity $\eta(t)$ is given: $$ \eta(t) = \frac{V}{k_B T} \langle \sigma(t) \sigma(0) \rangle, $$ identified as ...
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1answer
203 views

Calculating density of states given energy levels and degeneracy

In my statistical mechanics class, my professors did a problem in which he calculated the density of states, however I am having trouble justifying his approach. I did the problem beforehand in an ...
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0answers
34 views

Must the whole universe have the same entropic arrow of time?

Theoretically, could it be possible for one galaxy to have low entropy in the past, but another nearby to have low entropy in the future? I understand that, cosmologically, there almost certainly are ...
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2answers
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Deriving density of states in different dimensions in k space

The results for deriving the density of states in different dimensions is as follows: 3D: $g(k)dk = 1/(2\pi)^3 4 \pi k^2 dk$ 2D: $g(k)dk = 1/(2\pi)^2 2 \pi k dk$ 1D: $g(k)dk = 1/(2\pi) 2 dk$ I get ...
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1answer
52 views

Can Einstein relation be used to calculate mobility under equilibrium dynamics?

In weak field non-equilibrium dynamics, mobility can be calculated by Einstein relation $\mu=\frac{eD}{K_BT}$, where $D$ is diffusion constant. Mobility can also be calculated by the definition $\mu=...
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0answers
44 views

Why are thermodynamic potentials minimized?

In thermodynamics one says that in equilibrium the corresponding thermodynamic potential is minimized. Why? For example take the case of a canonical ensemble. Based on the assumption that the ...
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0answers
55 views

How energy $E= \pi k T$? [duplicate]

According to the Equipartition Theorem, $$E_{kin}= 3/2 k_B T$$ I read that when we are keeping the wave-nature of particles in mind, we can write $$E= \pi k_B T$$ But how we can write that?
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26 views

Order parameter and Bose-Einstein condensation

I want to study about order parameter and symmetry breaking related to bose einstein condensation in interacted system.which book i should read.also i want to learn this in second quantization ...
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2answers
82 views

Does quantum gases obey ideal gas equation $ PV= nRT$?

At extremely low temperature, does an ideal gas of bosons or fermions obey the ideal gas equation, $PV= nRT$?
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Deviation from ideal gas [closed]

Can anybody help me solving this problem ? I am trying to solve this one by myself.
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1answer
2k views

Math needed for undergrad Statistical Mechanics/Thermal Physics [duplicate]

A professor recommended me to take a course on Statistical Physics as preparation for agent-based computing in social sciences. Now I have no experience in physics beyond basic highschool, and ...
2
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1answer
6k views

Why liquids and solids are mostly regarded as incompressible?

In many continuum-mechanical Problems it is assumed that liquid and solid substances cannot Change the total value of volume where it holds $\rho = const, \vec{\nabla}\cdot \vec{v} = 0$. In the 1-...
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2answers
208 views

Unphysical region of Helmholtz free energy for Van der Waals gas

Recently, while looking at the Van der Waals gas and its implications for phase transitions I stumbled across a problem. We derived the normal Van der Waals gas equation: $$ \left(p+\frac{aN^2}{V^2}\...
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0answers
14 views

Aggregation phenomena : How to get from a discrete to a continuous point of view

I'm studying a diffusion limited aggregation phenomenon. The $N$ particles diffuse in a box and when there is a contact they stick with a probability $p$, and let's say to simplify $p=1$. Meaning that ...
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1answer
209 views

Deriving the Boltzmann Distribution

I have a basic understanding of thermodynamics, and I came across this derivation of the Boltzmann distribution. I understand all of it except the end and I need some clarification. At the end, the ...
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0answers
31 views

What is the equation of state of foam, on a macroscopic scale?

Consider a large amount of soap foam (or any other substance producing foam), of mass density $\rho$ in a gravityless environment. What is the internal pressure $p$ of that foam, viewed as a fluid on ...
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0answers
50 views

Wolff cluster update in Monte Carlo simulation - at critical temperature [closed]

A general question to the Monte Carlo experts. When I use Wolff algorithm for global updates, say for Ising 2d, I always flip at least one spin (the initial random spin in the cluster). So, near the ...
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7answers
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Is there a classical analog to quantum mechanical tunneling?

In comments to a Phys.SE question, it has been written: 'Tunneling' is perfectly real, even in classical physics. [...] For sufficiently large temperatures this can put the system above a hump in ...
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2answers
52 views

Kinetic Theory of Solids

Is there a clean way to examine temperature for solids and liquids in classical mechanics like the kinetic theory for gases? I'd like to get a good explanation that doesn't involve much in the way of ...
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1answer
66 views

How to explain imaginary kinematic viscosity of a vacuum?

According to the connection between the Schrödinger equation and the Navier-Stokes vacuum has the imaginary kinematic viscosity $\frac{ih}{2m}$. How to explain it? For the formation of the viscosity ...
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1answer
124 views

Interpretation of the Boltzmann factor and partition function

$$p_i = \frac{ \exp\left(-\frac{\epsilon _i}{k_BT} \right)}{Z} $$ $$ Z= \sum_{i} \exp\left(-\frac{\epsilon _i}{k_BT} \right)$$ A) Is $p_i$ the probability of the system having an energy equal to $\...
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2answers
56 views

Mermin Wagner theorem proof, what does the K stand for ?

I've been reading about the Mermin-Wagner theorem recently. I think I understand pretty much every computation need to derive its result from the Bogoliub inequality, but there is one thing I don't ...